# Partial fraction decomposition

### Partial fraction decomposition

#### Lessons

$\bullet$ Partial fraction decomposition expresses a rational function $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomials in $x$, as a sum of simpler fractions.

$\bullet$Partial fraction decomposition only applies to proper fractions in which the degree of the numerator is less than that of the denominator.

• Introduction
Introduction to Partial Fraction Decomposition
a)
What is partial fraction decomposition?

b)
When can we perform partial fraction decomposition?

• 1.
Case 1: Denominator is a product of linear factors with no repeats

Find the partial fractions of:

a)
$\frac{x + 7}{(x + 3)(x - 1)}$

b)
$\frac{4x + 3}{x^{2} + x}$

• 2.
Case 2: Denominator is a product of linear factors with repeats

Find the partial fractions of :

a)
$\frac{3x^{2} - 5}{(x - 2)^{3}}$

b)
$\frac{2x - 1}{x^{2} + 10x + 25}$

• 3.
Case 3: Denominator contains irreducible quadratic factors with no repeats

Find the partial fractions of :

$\frac{2x^{2} + 5x + 8}{x^{3} - 8x}$

• 4.
Case 4: Denominator contains irreducible quadratic factors with repeats

Find the partial fractions of:

$\frac{3x^{4} + x^{3} + 1}{x(x^{2} + 1)^{2}}$

• 5.
First perform long division, then partial fraction decomposition

Find the partial fractions of:

a)
$\frac{x^{3} - 3x^{2} + 4x}{x^{2} - 3x 2}$

b)
$\frac{2x^{2} + 14x + 24}{x^{2} + 6x - 16}$